1-introductiontostressandstrain-120127223145-phpapp02

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MechanicsofFlexibleMaterials

ByHammad Mohsin

1

CourseOutlineA)FUNDAMENTALS&POLYMERS

Module1IntroductiontoMechanicsofMaterials

,

StressesandDeformations,TrueStressandTrue

Strain

Module2StudyofStressandStrain

Stress StrainDiagramsofDuctileandBrittle

Materials,IsotropicandAnisotropicMaterials,

ModulusofElasticity,ModulusofRigidity,Elasticand

PlasticBehaviorofMaterials,NonLinearElasticity,

LinearElasticity,2


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StressandStraininChangedThermalConditions,

RepeatedLoading,BendingofElastoplastic

Materials,Analysis

ofStresses

and

Deformations

Module3MolecularbasisofRubberlikeelasticity

StructureofaTypicalNetwork, Elementary

MolecularTheories,MoreAdvancedMolecular

TheoriesPhenomenologicalTheoriesand

MolecularStructure,SwellingofNetworksand

ResponsiveGelsEnthalpicandEntropic

ContributionstoRubberElasticity:Force

TemperatureRelations, DirectDeterminationofMolecularDimensions

3

Module4StrengthofElastomers

InitiationofFracture,ThresholdStrengthsandExtensibilities,FractureUnderMultiaxial

Stresses,CrackPropagation, TensileRupture,

epea e ress ng: ec an ca a gue,

SurfaceCrackingbyOzone,AbrasiveWear.

Module5FailurePrevention

,aidsforpreventingbrittlefailure,Defect

analysisHDPEpipedurability.

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B)TEXTILEMATERIALS

Module6MechanicalProperties ofTextileFibres

TensileRecovery,ElasticPerformanceCoefficientin

Tension,InterFibreStressanditsTransmission,

Stressanalysisofstablefibre,filaments,influence

oftwistonyarnmodulus

Plasticityoftextilefibersbasedoneffectofload,

time,temperaturesuperposition.

Module7MechanicsofYarns:

MechanicsofBentYarns,FlexuralRigidity,FabricWrinkling,StiffnessinTextileFabrics.Creasingand

CreaseproofingofTextiles5

Module8CompressionofTextileMaterials

StudyofResilience,FrictionbetweenSingle

Fibres,FrictioninPliedYarns

Module9MechanicalPropertiesofNon

Wovensandcompositematerials

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Books

Neilsen

L.,

Landel

R.,Mechanical

Properties

of

Polymers

andComposites(1994)

MarkJ,ErmanB.,ElrichFScienceandTechnologyof

Rubbers(2005)

FerdinandPBeer,ERussellJhonstonJr.,JhonTDewolf

MechanicsofMaterials(2004)

(2004)

AEBogdanovich,CMPastoreMechanicsofTextileand

LaminatedComposites(1996)

7

Assessment

Quizzes: 10%

Classparticipation &Discussion 10%

Assignments: 10%

Midterm: 30%

Final: 40%

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Cause

&effect

model

9

MaterialProperties

1. Elastic

a. Elastic Behavior causes a

materialtoreturntoits

originalshapeafterbeing

deformed.

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b. Completelyelasticbehavior

Force

(F)

kxF =k is called the elastic

modulus

Distance (x)

11

2.Viscousa. Viscousbehaviorisrelatedtotherateof

e orma on.

t

xF

Viscosity Rate of deformation

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force, F

as

slow

distance, x

13

3.

Viscoelastic

a. Fibersexhibitviscoelasticbehavior

b.forcerequiredtodeformamaterial

dependentsamountofdeformationand rate

atwhichthematerialisdeformed

fast

x

elastic

slow

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B. InternalStructure1. ChemicalComposition

2. Crystallinity

Polymerchainsorsectionspackedtogether

3. Orientation

Alignmentofchainsalongfiberaxis

.MeltingTemperature

2. GlassTransitionTemperature

Mostpolymersarethermoplastic they soften

beforemelting15

D.PhysicalProperties

BreakingStrength

Force re uired to break a fiber

2. BreakingElongation

Amountofstretchbeforebreaking

3. Modulus

Resistancetodeformation

4. Toughness

Amountofenergyabsorbed

5. Elasticity

Abilitytorecoverafterbeingdeformed

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17

Structuralfactors=>Mechanical

Behavior

l.Molecularweight

.

3.Crystallinity andcrystalmorphology

4.Copolymerization(random,block,andgraft)

5.Plasticization

6.Molecularorientation

.

8.Blending

9.Phaseseparationandorientationinblocks,grafts,andblends

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ExternalFactorsMechanical

Properties

1.Temperature

2.Time,frequency,rateofstressingorstraining3.Pressure

4.Stressandstrainamplitude

5.Typeofdeformation(shear,tensile,biaxial,etc.)

. ea rea men sor erma s ory

7.Nature

ofsurrounding

atmosphere,

especially

moisturecontent

19

5assumptions >MechanicalBehavior

1)Linearity:Twotypesoflinearityarenormallyassumed:A Materiallinearit Hookean stressstrainbehavior)orlinearrelationbetweenstressandstrain;B)Geometriclinearityorsmallstrainsanddeformation.

2)Elastic:Deformationsduetoexternalloadsarecompletelyandinstantaneouslyreversibleuponloadremoval.

3)Continuum:Matteriscontinuouslydistributedforallsizescales,i.e.therearenoholesorvoids.

4)Homogeneous:Materialpropertiesarethesameateverypointormaterialpropertiesareinvariantupontranslation. 20


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5)Isotropic:Materialswhichhavethesame

mec an ca propert es na rect onsatan

arbitrarypointormaterialswhoseproperties

areinvariantuponrotationofaxesatapoint.

Amorphousmaterialsareisotropic.

21

Stress Strain>Definations

DogBoneisusedandmaterialpropertiessuchas

1)Youngsmodulus,2)Poissonsratio,3)failure(yield)stressandstrain.

Thespecimenmaybecutfromathinflatplateofconstantthicknessormaybemachinedfromacylindricalbar.

Thedogboneshapeistoavoidstressconcentrationsfromloadingmachineconnectionsandtoinsureahomogeneousstateofstressandstrainwithinthemeasurementregion.

Thetermhomogeneoushereindicatesauniformstateofstressorstrainoverthemeasurementregion,i.e.thethroatorreducedcentralportionofthespecimen.

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Theengineering(average)stresscanbe

ca cu ate y v ngt eapp e tens e

force,P,(normaltothecrosssection)bythe

areaoftheoriginalcrosssectionalareaA0 as

follows,

Stress

23

Strain Theengineering(average)straininthedirection

ofthetensileloadcanbefoundbydividingthechangeinlength,L,oftheinscribedrectanglebytheoriginallengthL0,

Theterm lambdaintheaboveequationiscalledtheextensionratioandissometimesusedforlargedeformationse.g.,Lowmodulusrubber

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TrueStressesandStrain

Truestressandstrainarecalculatedusingthe

nstantaneous e orme atapart cu ar oa

valuesofthecrosssectionalarea,A,andthe

lengthof therectangle,L,

25

YoungModulus

Youngsmodulus,E,maybedeterminedfrom

es opeo es resss ra ncurveor y

dividingstressbystrain,

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theaxialdeformationoverlengthL0is,

Poissonsratio,,isdefinedastheabsolute

valueoftheratioofstraintransverse, y,to

theloaddirectiontothestrainintheload

direction, x ,

Wherestraintransverse

veforAppliedtensileload, 27

Shear

L=lengthofthecylinder,

T=appliedtorque,

r=radial distance,

J=polarsecondmomentofarea

G=shear modulus.

=shearstress, =angleoftwist,

=shearstrain, 28


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Theshearmodulus,G,istheslopeoftheshear

stressstraincurveandmaybefoundfrom,

wheretheshearstrainiseasilyfoundbymeasuring

onlytheangularrotation,,inagivenlength,L.

TheshearmodulusisrelatedtoYoungsmodulus

AsPoissonsratio,,variesbetween0.3and0.5for

mostmaterials,the shearmodulusisoften

approximatedby,G~E/3. 29

TypicalStressStrainProperties

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Yieldpoint ifthestressexceedstheproportionallimita

residualorpermanentdeformationmayremain

whenthes ecimenisunloadedandthematerialis

saidtohaveyielded.

Theexactyieldpointmaynotbethesameasthe

proportionallimitandifthisisthecasethelocation

isdifficulttodetermine.

Asaresult,anarbitrary0.2%offsetprocedureis

oftenusedtodeterminetheyieldpointinmetals

31

Thatis,alineparalleltotheinitialtangentto

es resss ra n agram s rawn opass

throughastrainof0.002in./in.

TheyieldpointisthendefinedasthepointC

ofintersectionofthislineandthestressstrain

dia ram.

Thisprocedurecanbeusedforpolymersbut

theoffsetmustbemuchlargerthan0.2%

definitionusedformetals.

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thestressisnearly

linearwithstrainuntilitreachesthe

upperyieldpoint

stresswhichisalso

knownasthe

elasticplastic

tensileinstability

point.

Atthispointtheload(orstress)decreasesasthe

deformationcontinuestoincrease.Thatis,lessload

isnecessarytosustaincontinueddeformation.

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Theregionbetweentheloweryieldpoint

andthemaximumstressisaregionofstrain

hardening,PolyCarbonateshowsthesimilar

behavior

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IfthestrainscaleofFig.(a)isexpandedas

illustratedinFig.(b),

thestressstraindiagramofmildsteelis

approximatedbytwostraightlines;

i)forthelinearelasticportionand

ii)ishorizontalatastresslevelofthelower

yieldpoint. 35

Thischaracteristicofmildsteeltoflow,

nec or raw w ou rup urew en e

yieldpointhasbeenexceededhasledtothe

conceptsofplastic,limitorultimatedesign.

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IdealizedStress Strain

alinearelasticperfectlybrittlematerialisassumed

tohaveastressstraindiagramfig(a)

aperfectlyelasticplastic materialwiththestress

straindiagramFig(b)mildsteelorPolyC 37

Metals(andpolymers)oftenhavenonlinear

stressstrainbehaviorasshowninFig.(a).Thesearesometimesmodeledwithabilineardiagram

asshowninFig.(b)andarereferredtoasa

perfectlylinearelasticstrainhardeningmaterial.38


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2

MathematicalDefinitions

DefinitionofaContinuum:Abasicassumption

o e ementaryso mec an cs st ata

materialcanbeapproximatedasacontinuum.

Thatis,thematerial(ofmassM)is

continuouslydistributedoveranarbitrarily

smallvolume,V,suchthat,

39

Mathematical/Physical Def.of

NormalandShearStress

Considerabodyin

equ r umun er e

actionofexternal

forces

F1,F2,F3,F4=Fi as

showninFi .

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2

Ifacuttingplaneis

passe t roug t e

bodyas

showninFig,

equilibriumis

maintainedonthe

remainingportion

byinternalforces

distributedoverthe

surfaceS. 41

Atanyarbitrarypointp,

the incrementalresultantforce, Fr,onthe

cutsurfacecanbebrokenupintoanormal

forceinthedirectionofthenormal,n,to

surfaceSand

atangentialforceparalleltosurfaceS.

Thenormalstressandtheshearstressat

po ntp smat emat ca y e ne as,

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2

Alternatively,theresultant

force,Fr,atpointpcan

bedivided

by

the

area,

A,

an e m a en o

obtainthestressresultant

rasshowninFig.Normal

andtangential

componentsofthisstress

resultantwill thenbethe

normalstress nandshearstress satpointp

ontheareaA.43

Ifapairofcuttingplanesadifferentialdistance

apartarepassedthrough

thebod arallel toeachofthethreecoordinate

planes,acubewillbeidentified.

Eachplanewillhavenormalandtangential

componentsofthestressresultants.

Thetangentialorshearstressresultantoneach

planecanfurtherberepresentedbytwocomponentsinthecoordinatedirections.

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2

Theinternalstress

stateisthen

representedby

threestress

eachcoordinate

planeasshownin

Fig.Thereforeat

anypointinabody

t erew en ne

stresscomponents.

Theseareoften

identifiedinmatrix

formsuchthat, 45

Usingequilibrium,itiseasytoshowthatthe

s ressma r x ssymme r c,

or

leavingonlysixindependentstressesexistingat

amaterialpoint.46


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2

PhysicalandMathematicalDef.of

Normal

&

Shear

Strain

Ifthereisstressactingonthebody.For

examp e

47

Bothshearingandnormaldeformationmayoccur

withdisplacements. uisthedisplacementcomponentinthex

directionandvisthedisplacementcomponentin

ey rec on.

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2

Theunitchangeinthexdimensionwillbethe

strain xxandisgivenby,

Ifweapplysimilarlyforyandzdirection,and

assumethatchangeofangleisverysmallthen

uwillbeignored.Thenin3coordinate

systemnormalstrainsaredefinedas:

49

Shearstrains Shearstrainsaredefinedasthedistortionof

theoriginal90angleattheoriginorthesum

oftheangles 1+2.Thatis,againusingthe

smalldeformationassumption,

Aftersolvinginall3directionsshearstrainis

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Likestresses,ninecomponentsofstrainexist

atapointandthesecanberepresentedin

matrixformas,

A ain itis ossibletoshowthatthestrain

matrixissymmetricorthat,

Hencethereareonlysixindependentstrains. 51

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